Chapter 9.2, Problem 57E

Insurance. An auto insurance company classifies its customers in three categories: poor, satisfactory, and preferred. Each year, $40\%$ of those in the poor category are moved to satisfactory, and $20\%$ of those in the satisfactory category are moved to preferred. Also, $20\%$ in the preferred category are moved to the satisfactory category, and $20\%$ in the satisfactory category are moved to the poor category. Customers are never moved from poor to preferred, or conversely, in a single year. Assuming that these percentages remain valid over a long period of time, how many customers are expected in each category in the long run?